# Virtual Math Learning Center

## NEW! Section 6.4 – Work

### Instructions

• The first set of videos below explain the concepts in this section.
• This page also includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
• You can find additional practice problems for this section on the Practice Problems page for this Module.

### Concepts

• Using integration to calculate work
• Using Hooke’s Law to find the work done when stretching a spring and other application problems involving work and springs
• Finding the work done lifting a rope with a weight at the end
• Finding the work to pump water out of a tank

If you would like to see more videos on the topic, click the following link and check the related videos.

If you would like to see more videos on the topic, click the following link and check the related videos.

If you would like to see more videos on the topic, click the following link and check the related videos.

If you would like to see more videos on the topic, click the following link and check the related videos.

### Exercises

Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.

1. How much work is done in raising a 60 kg barbell from the floor to a height of 2 meters?

$$1176$$ Joules

If you would like to see more videos on this topic, click the following link and check the related videos.

2. How much work is done in lifting a 50 pound weight from a height of 6 inches to a height of 18 inches?

$$50$$ ft-lbs

If you would like to see more videos on this topic, click the following link and check the related videos.

3. When a particle is at a distance $$x$$ meters from the origin, a force of $$f(x)=2x^2+1$$ Newtons acts on the object, thus causing the object to move horizontally along the $$x$$-axis. How much work is done in moving the object from $$x=1$$ to $$x=2?$$

$$\displaystyle \frac{17}{3}$$ J

If you would like to see more videos on this topic, click the following link and check the related videos.

4. A spring has a natural length of 1 m. If a 25-N force is required to keep it stretched to a length of 3 m, how much work is done in stretching the spring from 2 m to 5 m?

$$\displaystyle \frac{375}{4}$$ J

If you would like to see more videos on this topic, click the following link and check the related videos.

5. Suppose the work required to stretch a spring 1 foot beyond its natural length is 12 foot pounds.

1. How much work is needed to stretch it 9 inches beyond its natural length?
2. How far beyond its natural length will a force of 16 pounds keep it stretched?

1. $$\dfrac{27}{4}$$ ft-lbs
2. $$\dfrac{2}{3}$$ ft

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6. A tank is in the shape of an upright cylinder with radius 5 feet and height 20 feet. The tank is half filled with water. Find the work required to pump the water to the top of the tank. Clearly indicate where you are placing the axis and which direction is positive.

$$234,375\pi$$ ft-lbs

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7. A rectangular tank 10 m long, 8 m wide and 12 m is deep filled with water. Find the work required to pump the top 5 m of water out the top of the tank. Clearly indicate where you are placing the axis and which direction is positive.

$$\displaystyle W=\int_0^5 (80)(9800)y\, dy = 9,800,000$$ J

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8. A trough is in the shape of a half drum (half a cylinder lying on its side). The length of the tank is 10 meters and the radius is 4 meters. Assuming it is full of water, find the work done in pumping the water to the top of the tank. Clearly indicate where you are placing the axis and which direction is positive.

$$\dfrac{20}{3}(9800)(64)$$ J $$\quad$$ or $$\quad \dfrac{12,544,000}{3}$$J

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9. A triangular trough has a length of 5 feet, a distance of 2 feet across the top and a height of 3 ft. Assuming it is full of water, set up but do not evaluate an integral that gives the work done in pumping the water through a spout located at the top of the trough with length 0.5 feet. Clearly indicate where you are placing the axis and which direction is positive.

$$\displaystyle W=\frac{10}{3} \rho g \int_0^3 y\left( \frac{7}{2}-y\right) \, dy$$

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10. A tank is in the shape of a cone with radius $$r=3$$ feet and height $$h=8$$ feet. Assuming it is full of water, set up but do not evaluate an integral that gives the work it takes to pump the water out of the tank. Clearly indicate where you are placing the axis and which direction is positive.

$$\displaystyle W=\frac{9\pi}{64}\rho g \int_0^8 y^2(8-y) \, dy$$

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11. A 100 foot long rope that weighs $$\displaystyle{{1}\over{5}}$$ pounds per foot hangs vertically from the top of a tall building.  Find the work done in pulling the entire rope to the top of the building.

$$1,000$$ ft-lbs

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12. A 500 foot rope that weighs 3 pounds per foot is used to lift an 80 pound weight up the side of a building that is 625 feet tall. Find the work done.

$$415,000$$ ft-lbs

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13. A 200 pound cable is 100 feet long and hangs vertically from the top of a tall building. How much work is done in pulling the first 10 feet of the cable to the top of the building?

$$1,900$$ ft-lbs

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